let P be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis:
let s be State of SCM+FSA; :: thesis:
let I be Program of SCM+FSA; :: thesis:
let a be read-write Int-Location ; :: thesis:
set s0 = Initialized s;
set IA = Initialize I;
assume A1: I is_closed_onInit s,P ; :: thesis:
A2: ProgramPart I = I by RELAT_1:209;

let k be Element of NAT ; :: thesis:
s +* (Initialized I) = (Initialized s) +* (Initialize I) by SCMFSA8A:13;
hence IC (Comput ((P +* I),((Initialized s) +* (Initialize I)),k)) in dom I by A1, SCM_HALT:def 4; :: thesis:

end;

then A3: I is_closed_on Initialized s,P by SCMFSA7B:def 7, A2;
assume I is_halting_onInit s,P ; :: thesis:
then A4: P +* I halts_on s +* (Initialized I) by SCM_HALT:def 5;
s +* (Initialized I) = (Initialized s) +* (Initialize I) by SCMFSA8A:13;
then A5: I is_halting_on Initialized s,P by A4, SCMFSA7B:def 8, A2;
assume s . a > 0 ; :: thesis:
then A6: (Initialized s) . a > 0 by SCMFSA6C:3;

hereby :: thesis:

let k be Element of NAT ; :: thesis:
A7: s +* (Initialized (while>0 (a,I))) = (Initialized s) +* (Initialize (while>0 (a,I))) by SCMFSA8A:13;
A8: s +* (Initialized I) = (Initialized s) +* (Initialize I) by SCMFSA8A:13;
assume k <= (LifeSpan ((P +* I),(s +* (Initialized I)))) + 3 ; :: thesis:
hence IC (Comput ((P +* (while>0 (a,I))),(s +* (Initialized (while>0 (a,I)))),k)) in dom (while>0 (a,I)) by A7, A3, A5, A6, A8, SCMFSA_9:47; :: thesis:

end;